new controllers efficient model based design method

Industrial Engineering Letters www.iiste.org

ISSN 2224-6096 (Paper) ISSN 2225-0581 (online)

Vol.3, No.7, 2013

1

New Controllers Efficient Model-Based Design Method

Farhan A. Salem1,2

(Corresponding author) 1Mechatronics Program. Dept. of Mechanical Engineering, Faculty of Engineering, Taif University, 888, Taif,

Saudi Arabia, 2Alpha Center for Engineering Studies and Technology Researches (ACESATR), Amman, Jordan.

Email: [email protected]

Abstract

This paper proposes a new simple and efficient model-based time domain P, PI ,PD , and PID controllers design

methods for achieving an important design compromise; acceptable stability, and medium fastness of response,

the proposed method is based on selecting controllers' gains based on plant's parameters, a simple expressions

are proposed for calculating and soft tuning controller's gain, the proposed controllers design methods were

tested for first, second and first order system with time delay, and using MATLAB/simulink software.

Keywords: Controller, controller design.

1. Introduction The term control system design refers to the process of selecting feedback gains (poles and zeros) that meet

design specifications in a closed-loop control system. Most design methods are iterative, combining parameter

selection with analysis, simulation, and insight into the dynamics of the plant (Ahmad A. Mahfouz, et al 2013).

An important compromise for control system design is to result in acceptable stability, and medium fastness of

response, one definition of acceptable stability is when the undershoot that follows the first overshoot of the

response is small, or barely observable. Beside world wide known and applied controllers design method

including Ziegler–Nichols, Chiein-Hrones-Reswick (CHR), Wang–Juang–Chan, Cohen-Coon, many controllers

design methods have been proposed in different papers and texts including (Astrom K,J et al 1994)( Ashish

Tewari, 2002 )( Katsuhiko Ogata, 2010)( Norman S. Nise, 2011)( Gene F. Franklin, et al 2002)( Dale E. Seborg,

et al, 2004)( Dingyu Xue et al, 2007)( Chen C.L et al, 1989)( R. Matousek, 2012)( K. J. Astrom et al, 2001)(

Susmita Das et al, 2012) (L. Ntogramatzidis, 2010)( M.Saranya et al, 2012 ), each method has its advantages,

and limitations. (R. Matousek, 2012 ) present multi-criterion optimization of PID controller by means of soft

computing optimization method HC12. (K. J. Astrom et al, 2001) introduce an improved PID tuning approach

using traditional Ziegler-Nichols tuning method with the help of simulation aspects and new built in function. (L.

Ntogramatzidis et al, 2010) A unified approach has been presented that enable the parameters of PID, PI and PD

controllers (with corresponding approximations of the derivative action when needed) to be computed in finite

terms given appropriate specifications expressed in terms of steady-state performance, phase/gain margins and

gain crossover frequency. (M.Saranya et al, 2012) proposed an Internal Model Control (IMC) tuned PID

controller method for the DC motor for robust operation. (Fernando G. Martons, 2005 ) proposed a procedure

for tuning PID controllers with simulink and MATLAB. (Saeed Tavakoli, 2003) presented Using dimensional

analysis and numerical optimization techniques, an optimal method for tuning PID controllers for first order plus

time delay systems.

This paper proposes P, PI, PD, and PID controller design method based on selecting controller gainsbased on

plant's parameters that meet an important design compromise; acceptable stability, and medium fastness of

response. By relating controller's gains and plant parameters, particularly, time constant, damping ratio and

undamped natural frequency, expressions for selecting values of controllers' gains are to be derived for FOPTD,

first and second order systems, as well as, systems that can be approximated as first and second order systems, to

achieve more smooth response with minimum overshoot, minimum settling time, and minimum steady state

error, a soft tuning parameters with recommended ranges are to be introduced.

1.1 Controllers Modeling

The controller that will be considered are Proportional, proportional derivative, proportional integral and

proportional integral derivative controllers

Proportional Control: The control action of P-controller is proportional to the error, The relation between the

output control signal of controller, u(t) and the actuating error signal e(t) is given by Eq.(1), taking Laplace-

transform and manipulating Eq.(1), for transfer function gives:

( ) ( ) ( ) ( ) p 牋牋 Kpu t K e t U s E s= ⇒ = (1)

Gp(s) = U(s )/E(s) = Kp (2)

Proportional-Derivative, PD-controller: The output control signal of PD-Controller controller u(t),is equal to

the sum of two signals and given by Eq.(3), taking Laplace transform and solving for transfer function, gives



Vol.3, No.7, 2013

2

Eq.(4) :

( )( ) ( ) ( ) ( ) ( )

P D P D

de tu t K e t K U s K E s K sE s

dt= + ⇔ = + (3)

( ) ( ) ( )P

PD P D D D PD

D

KG s K K s K s K s Z

K= + = + = + (4)

Where: ZPD = KP/KD, is the PD-controller zero,

Proportional-Integral, PI-controller: The output control action signal u(t), of PI- controller is proportional to

the error and the integral of error, where the integral of the error, as well as, the error itself are used for control,

and given by Eq.(5), taking Laplace transform, and solving for transfer function gives Eq.(6):

1( ) ( ) ( ) ( ) ( ) ( ) ( ) I

P I P I P

Ku t K e t K de t dt U s K E s K E s E s K

s s

= + ⇔ = + = +

∫ (5)

( )( )

( )

I

P

I P I P P PI

PI P

KK s

K K s K K K s ZG s K

s s s s

++ +

= + = = = (6)

Where: /PI I PZ K K= , is the PI-controller zero. Equation (6) can be rewritten, in terms of integral time

constant TI, to have the following given by (7), and implemented as shown in Figure 15(b) :

1 1( ) (1 ) (1 ) ( ) ( ( ) ( ) )I I

PI P P P P

P I

K KG s K K K u t K e t e t dt

s K s T s Ti= + = + = + ⇔ = + ∫

( ) ( )P

eu t K e

Ti= + (7)

This means if constant error exists, the controller action will keep increasing, until the error is zero, where:

TI=KP/KI, is the time constants of the integral actions, or integral time .

Proportional-Integral- Derivative PID-controller: Combining all three controllers, results in the PID

controller, the output of PID controller is equal to the sum of three signals and given by Eq.(32), taking Laplace

transform, and solving for transfer function , gives Eq.(8)

( ) 1( ) ( ) ( ) ( ) ( ) ( ) ( )P D I P D I

de tu t K e t K K e t dt U s K E s K E s s K E s

dt s= + + ⇔ = + +∫

( ) ( ) ( )I IP D PID P D

K KU s E s K K s G s K K s

s s

= + + ⇒ = + +

(8)

This equation can be manipulated to result in the form given by Eq.(9)

2

2

( )

P ID

D DI D P IPID P D

K KK s s

K KK K s K s KG s K K s

s s s

+ + + + = + + = = (9)

Equation (9) is second order system, with two zeros and one pole at origin, and can be expressed to have the

form given by Eq.(10), which indicates that PID transfer function is the product of transfer functions PI and PD ,

Implementing these two controllers jointly and independently will take care of both controller design

requirements

( )( ) ( ) ( )( ) ( )

D PI PD PD

PID D PI PD PI

K s Z s Z s ZG K s Z G s G s

s s

+ + += = + = (10)

The transfer function of PID controller, also ,can also be expressed to have the form given by Eq.(11):

( ) ( ) ( )2 ( )D PI PD D PI PD D PI PD D

PID

K s Z s Z K s Z Z K s Z Z KG

s s

+ + + + += = (11)

Rearranging Eq.(11), we have:

( ) ( )2 ( ) ( )PI PD DD PI PD D PI PD D

PID PI PD D D

Z Z K sK s Z Z K Z Z KG Z Z K K s

s s s s

+= + + = + + +

Substituting the following values, ( )1 2 3,牋? ),牋牋PI PD D PI PD D DK Z Z K K Z Z K K K= + = = , gives:

2

1 3PID

KG K K s

s= + + (12)

Since PID transfer function is a second order system, it can be expressed in terms of damping ratio and

undamped natural frequency to have the form given by Eq.(13)::



Vol.3, No.7, 2013

3

22 22

( )

P I

D

D n nD D

PID

K KK s s

K s sK KG s

s s

ξω ω

+ + + + = = (13)

Where: 2 In

D

K

Kω = and 2 P

n

D

K

Kξω =

The transfer function of PID control given by Eq.(8) can, also, be expressed in terms of derivative time and

integral time to have the form given by Eq.(14): 2 11

1 I D IPID P D P

I I

T T s T sG K T s K

T s T s

+ += + + =

(14)

Where: TI= KP/ KI : integral time, TD= KD/ KP : derivative time,

/I P IK K T= , D P DK K T=

Since in Eq. (14) the numerator has a higher degree than the denominator, the transfer function is not causal and

can not be realized, therefore this PID controller is modified through the addition of a lag to the derivative term,

to have the following form:

D

11 ,牋牋牋 /N - time constant of the added lag

1

D

PID P

DI

T sG K

T sT s

N

= + + +

Where: N: determines the gain KHF of the PID controller in the high frequency range, the gain KHF must be

limited because measurement noise signal often contains high frequency components and its amplification

should be limited. Usually, the divisor N is chosen in the range 2 to 20. If no D-controller, then we have PI

controller, given by Eq. (15), it is clear that, PI and PD controllers are special cases of the PID controller.

111 I

PI P P

I I

T sG K K

T s T s

+= + =

(15)

The addition of the proportional and derivative components effectively predicts the error value at TD seconds (or

samples) in the future, assuming that the loop control remains unchanged. The integral component adjusts the

error value to compensate for the sum of all past errors, with the intention of completely eliminating them in TI

seconds (or samples). The resulting compensated single error value is scaled by the single gain KP.

1.2 Dominant features Most complex systems have dominant features that typically can be approximated by either a first or second

order system response. Control system's response is largely dictated by those poles that are the closest to the

imaginary axis, i.e. the poles that have the smallest real part magnitudes, such poles are called the dominant

poles, many times, it is possible to identify a single pole, or a pair of poles, as the dominant poles. In such cases,

a fair idea of the control system's performance can be obtained from the damping ratio and undamped natural

frequency of the dominant poles (Farhan A. Salem, 2013). For complex system, the controller gains are to be

selected based on plant's parameters (time constant, the damping ratio and undamped natural frequency) of the

dominant poles.

2. Controllers design for first order systems

First order systems and systems that can be approximated as first order systems, are characterized, mainly,

by time constant T. Time constant is a characteristic time that is used as a measure of speed of response to a step

input and governs the approach to a steady-state value after a long time. The general form of first order system's

transfer function in terms of time constant T, is given by Eq.(16).

( ) 1G s 牋

1Ts=

+ (16)

2.1 P-controller design for first order systems To design P-controller, with minim settling time, the proportional gain KP, is set equals to time constant, but

since different step input values can be applied, and correspondingly different steady state value will results, then

by multiplying plant's time constant T, by desired output value ( reference input) the value of KP can be selected

to achieve suitable response. To soften the response in terms of reducing overshoot, settling time and steady state

error, a tuning parameters α can be introduced, based on all this expression given by Eq.(17) can be proposed for

designing P-controller for first order systems:

牋牋 PK RTα= (17)

Where: T : Plant's time constant. R : desired output value, α: a tuning parameter that takes the value from 1 to

100. To achieve overall system fast response, with minimum overshoot and oscillation, parameter α, is softly

increased.



Vol.3, No.7, 2013

4

For systems with small DC gain and/or small time constant, to minimize time of design and selection process of

proportional gain KP, the expression given by Eq.(18) can be applied directly, where each increase of parameter

α will increase proportional gain by 100 factor

To test and verify the proposed P-controller design expression for selecting proportional gain KP for achieving

desired output of 10 with minimum possible overshoot, steady state error and settling time, four different first

order systems with different time constants, with transfer functions given by Eq.(19) and unity feedback control

systems simulink model shown in Figure 1 are to be used. The result of designing P-controller and tuned values

of parameter α, for first three systems, are shown in Table 1, the response curves are shown in Figure 2, these

response curves show that, acceptable stability, and medium fastness of response with minimum steady state

error can be achieved by applying proposed expression, and by selecting and increasing the value of only one

parameter α.

To test expression given by Eq.(18), applying it to system 3, and comparing with original expression, (see figure

2(c)(d)), the comparison shows that applying Eq.(18) to systems with small DC gain and/or small time constant,

simplifies and accelerate design process while achieving optimal response.

牋牋 10PK RTα= (18)

1 2 3 4

10 1 0.005 0.1( ) ,牋牋牋? ) ,牋牋牋牋( ) ?牋牋牋牋( ) ?

10 1 2 10 1 2 100sys sys sys sysG s G s G s G s

s s s s= = = =

+ + + + (19)

Figure 2 Simulink model for testing proposed design method

Table 1: P-controller design for first order system

P-Controller α System 1 System 2 System 3

T=1 T=0.5 T=10

P牋牋K RTα=

R=Vin=10

α=1 10 5 100

α=5 50 25 500

α=10 100 50 1000

Figure 2(a) P-controller design for system 1

0.1

2s+100

sys (3)1

0.005

10s+1

sys (3)

1

s+2

sys (2)

10

10s+10

sys (1)

output

1

Unity feedback

Step input

Vin

PID(s)

P, PI, PD, PID controllers

1

(Td/N)s+1

Fil ter

du/dt

=

Kp

.3

1

.2

sy13.mat

.1

..7

0

..5

..4

..,

0

..'1

0

..'

0

..

du/dt

.'

Kp

.

,.,.

,.,

,.'

Td

,,

1

s

,'2

,'1

,'

Kd

,

1/Ti

'1

'.

1

s

''

Ki

'

0 0.50

2

4

6

8

10

12

Time (seconds)

Magnitude

Tuned: α = 10

0 0.5 10

5

10

Time (seconds)

Magnitude

Sys1 P-design, α = 1

0 0.5 10

5

10

Time (seconds)

Magnitude

Tuned: α = 5



Vol.3, No.7, 2013

5

Figure 2(b) P-controller design for system 2

Figure 2(c) P-controller design for system 3 applying Eq.( 17).

Figure 2(d) P-controller design for system 3 applying Eq.( 18)

2.2 PI-controller design for first order systems

Based on plant's time constants, expression given by Eq.(20) are proposed, to design PI controller for first order

systems, these expression relate selecting PI controller gains values based, only, on plant's time constant . To

reduce overshoot and speed up response, soft tuning parameters α, is introduced.

To verify and test the proposed PI-controller design procedure, the same four first order systems are given in

Eq.(19) and simulink model shown in Figure 1, are used, where manual switches are used to switch control

system to PI controller. The result of designing PI-controller for achieving desired output of 10 with zero steady

state error and minimum settling time, as well as, tuned values of tuning parameter α, are shown in Table 2, the

response curves are shown in Figure 3, these response curves show, that a fast response, minimum overshoot and

with zero steady state error are achieved. To speed up response, tuning parameter α can be increased softly, this

is shown in Figure 3(b).

牋

P

P

I

PI

I

K T

K TK

T T

K TT T

K

α

αα

αα

=

= = =

= = =

(20)

0 0.50

2

4

6

8

10

12

(Sec)

Magnitude

Tuned: α = 10

0 0.5 10

5

10

(Sec)

Magnitude


0 0.5 10

5

10

(Sec)

Magnitude

Tuned: α = 5

0 500

5

10

Time (seconds)

Magnitude


0 500

5

10

Time (seconds)

Magnitude

Tuned: α = 5

0 5 100

5

10

Time (seconds)

Magnitude

Tuned: α = 550

0 5 10 150

5

10

(Sec)

Magnitude


0 5 100

5

10

(Sec)

Magnitude

Tuned: α = 5

0 50

5

10

(Sec)

Magnitude

Tuned: α = 10



Vol.3, No.7, 2013

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Table 2: PI-controller design for first order system

PI-Controller

Parameters System 1 System 2 System 3

T 1 0.5 10

α 1 1 1

KP 1 0.5 10

KI 1 1 10

TI 1 0.5 10

N 1 1 1

Figure 3(a) PI-controller design for system 1

Figure 3(b) PI-controller design for system 2

Figure 3(c) PI-controller design for system 3

2.3 PD-controller design for first order systems

Based on plants time constants, expressions given by Eq.(21) are proposed, to design PD controller for first

order systems. For first order systems with small time constant and/or small DC gain, expression given by

Eq.(22) are proposed, to reduce steady state error and speed up response, a soft tuning parameters α, is

introduced .To verify and test the proposed PD-controller design procedure, the same first order systems given in

Eq.(19) and simulink model shown in Figure 1 are used , manual switches are used to switch control system

components to PD controller. The result of designing PD-controller for achieving desired output of 10 with

minimum steady state error and minimum settling time, as well as, tuned values of tuning parameter alpha, are

shown in Table 3, the response curves are shown in Figure 4, the response curves show, that a fast response with

minimum steady state error are achieved. To speed up response and reduce steady state error tuning parameter α

are increased, this is shown in Figure 4. Since system 3, is with small both DC gain and time constant,

expressions given by Eq.(22) are applied, resulting in acceptable response compromise shown in Figure 4(b).

/ 1 /

P

D

D D P

K RT

TK

T K K R

α

αα

=

=

= =

(21)

0 5 100

5

10

Time (seconds)

Magnitude

Sys1 PI-design, α = 1

0 1 20

5

10

Time (seconds)

Magnitude

Tuned: α = 5

0 0.5 10

5

10

Time (seconds)

Magnitude

Tuned: α = 10

0 5 10 150

5

10

Time (seconds)

Magnitude


0 50

5

10

Time (seconds)

Magnitude

Tuned: α = 5

0 1 20

5

10

Time (seconds)

Magnitude

Tuned: α = 10

0 500 1000 15000

5

10

Time (seconds)

Magnitude


0 100 200 3000

5

10

Time (seconds)

Magnitude

Tuned: α = 5

0 10 200

5

10

Time (seconds)

Magnitude

Tuned: α = 100



Vol.3, No.7, 2013

7

2

2

/ 1 /

P

D

D D P

K R T

TK

T K K R

α

αα

=

=

= =

(22)

Table 3: Applying PD-controller design for first order system

PD-Controller

Parameters System 1 System 2 System 3

T 1 0.5 10

α 1 1 5

KP 10 5 5000

KD 1 .5 2

TD 0.1 0.1 4. e-004

N 1 1 1

Figure 4(a) PD-controller design for system 1, increasing tuning parameter alpha reduces error and speeds

response

Figure 4(b) PD-controller design for system 2,

Figure 4(b) PD-controller design for system 3, applying expression given by Eq.(22)

2.4 PID-controller design for first order systems

Based on plants time constants, expressions given by Eq.(23) are proposed, to design PID controller for first

order systems. Two tuning parameters are introduced α and ε, that can be tuned to speed up response, reduce

overshoot and steady state error, based on methods applied for derivation (calculations, analysis and trial and

error methods), a limits for these two parameters are proposed, these parameters are tuned softly, where: Tuning

parameter α :is responsible for reducing overshoot , (α =0.1:3), reducing will increase damping and

correspondingly, reduce overshoot, meanwhile, Tuning parameter ε : is responsible for speeding up response, (ε

=0.1:2)

0 0.50

5

10

Time (seconds)

Magnitude

Tuned: α = 5

0 1 20

5

10

Time (seconds)

Magnitude

Sys1 PD-design, α = 1

0 0.1 0.20

5

10

Time (seconds)

Magnitude

Tuned: α = 10

0 0.50

5

10

Time (seconds)

Magnitude

Tuned: α = 5

0 1 20

5

10

Time (seconds)

Magnitude


0 0.1 0.20

5

10

Time (seconds)

Magnitude

Tuned: α = 25

0 5 100

5

10

Time (seconds)

Magnitude

Tuned: α = 5

0 50

5

10

Time (seconds)

Magnitude


0 1 2 30

5

10

Time (seconds)

Magnitude

Tuned: α = 10



Vol.3, No.7, 2013

8

I

牋

* ,牋牋牋? 0.1 3

* ,牋牋牋? 0.1 2

*牋?

1T

*

P

I

D

D

D

P

P

I

K T

K T where

K T where

K TT

K T

K T

K T

α α

ε ε

εε

α α

=

= = ÷

= = ÷

= = =

= = =

(23)

To verify and test the proposed PID-controller design procedure, the same four first order systems are given in

Eq.( 19) and simulink model shown in Figure 1 are used , manual switches are used to switch control system to

PID controller. The result of designing PID-controller for achieving desired output of 10 with minimum steady

state error and minimum settling time, as well as, tuned values of tuning parameter alpha and epsilon, for system

1 are shown in Table 4, the response curves are shown in Figure 5, the response curves show, that a fast response

with minimum both overshoot and steady state error at less time are achieved.

Table 4 PD-controller design for system 1, responses are shown in Figure 5(a)

PD-Controller

System 1 System 1 System 1

T 1 1 1

α 3 1 0.5

ε 2 1 1

KP 1 1 1

KI 3 1 0.5

KD 2 1 1

TD 2 1 1

TI 0.333 1 2

N 1 1 1

Figure 5(a) PID controller design for system 1, and the effect of tuning parameters ε and α

Figure 5(b) PID controller design for systems 2 and 3,

2.5 Summary : Controller design for First order systems The proposed method and expressions for controllers terms selection and design for first order systems are

summarized in Table 5

0 5 100

5

10

15

Time (seconds)

Magnitude

Sys1 PID, α=1,ε=1

0 10 200

5

10

15

Time (seconds)

Magnitude

Sys1 PID, α=3,ε=2

0 5 10 150

5

10

15

Time (seconds)

Magnitude

Sys1 PID, α=0.5,ε=1

0 5 10 150

5

10

15

Time (seconds)

Magnitude


Magnitude

0 50 1000

5

10

15

Time (seconds)

Magnitude




Vol.3, No.7, 2013

9

Table 5 P, PI ,PD, PID controllers terms for first order systems

Controller

type

KP KI KD TD TI N

P-

controller

牋牋RTα

0

0

0

0

0

For sys.

small DC

gain and/or

small T

牋1? RTα

PD 牋牋RTα 0 牋? ?DK T α= /

1 /

D PK K

Rα

= 0 1 22÷

For sys.

small DC

gain and/or

small T

2R Tα 牋? ?DK T α= 21/ R α 0 0

PI

牋牋Tα

牋?

PK T

T T

α

α

=

=

0 0 T 1 22÷

PID T 牋* ,牋牋

0.1 3

Tαα = ÷

? ,

0.1 2

Tεε = ÷

*

0.1 2

D

D

P

KT

K

T

T

εε

ε

= =

=

= ÷

I

I

T

1T 牋

*

0.1 3

P

I

K

K

T

Tα αα

= =

= =

= ÷

1-22

3. Controllers design for second order systems

The general standard form of second order system, in terms of damping ratio ζ and undamped natural frequency

ωn is given by Eq.(24), knowing that that performance of second order systems depends on damping ratio ζ and

undamped natural frequency ωn , where damping ratio determines how much the system oscillates as the response

decays toward steady state and undamped natural frequency ωn, determines how fast the system oscillates

during any transient response (Farhan A. Salem, 2013), ωn has a direct effect on the rise time, delay time, and

settling time, therefore to speed up response and reduce ( remove) overshot, based on this an important design

compromise; acceptable stability, and medium fastness of response can be achieved by relating controller's gain

and plant's parameters 2

2 2( ) 牋牋

2

n

n n

G ss s

ωξω ω

=+ +

(24)

3.1 P-controller design for second order systems

Based on plant's damping ratio ζ and undamped natural frequency ωn, expression given by Eq.(25) is proposed

To design P-controller for second order systems. For systems with small time constant or DC gain expression

given by Eq.(26) is proposed. Only one tuning parameter α is introduced, that can be tuned to speed up response,

reduce overshoot, oscillation and steady state error and,

,牋牋牋牋牋牋牋牋牋牋牋0.1n

P

RK where is tuned from

α ωα

ξ= (25)

2

?牋牋牋牋牋牋牋牋牋牋牋5.n

P

RK where is tuned from

α ωα

ξ= (26)

Where R: is desired output value, α: can be increased for small R and decreased for large R. increasing

parameter α will result in increasing overshoot, oscillation and reducing steady state error.

To test and verify the proposed P-controller design, four second order systems are given by Eq.(27) and simulink

model shown in figure 6. Applying P-controller design for achieving desired output of 10, for three first



Vol.3, No.7, 2013

10

systems, will result in response curves shown in Figure 7, plants' parameters, tuned values of tuning parameter α

and ε, gains and resulted performance specifications, are shown in Table 6, the response curves show, system

will reach output with overshoot, oscillation and steady state error, increasing α will result in increasing

overshoot, reducing steady state error.

1 22 2

3 42 2

10 1? ) ,牋牋牋牋? ) ?牋牋

10 10 2 1

0.05 7? ) ,牋牋牋牋牋牋 ( )

2 2 1 2 5 1

sys sys

sys sys

G s G ss s s s

G s G ss s s s

= =+ + + +

= =+ + + +

(27)

Table 6 plants' parameters, tuned values of tuning parameter α and ε, gains and performance specifications

P-

Controller

Parameters System 1 System 1 System 2 System 3 System 4

ζ 0.5 0.5 0.3536 0.7071 2.5

ω 1 1 1.4141 0.7071 1

α 1 2 1 5 2

R 10 10 10 10 10

KP 20 40 199 500 8

TS 7.8 6.8 7 8 2

OS% 58% 73% 16.5% 16.5% 29%

Figure 6 Simulink model for verifying design

Figure 7(a) P-controller design for system 1, for α=1 , 2

7

s +5s+12

sys (3)1

0.05

2s +2s+12

sys (3)

1

s +s+22

sys (2)

10

10s +10s+102

sys (1)

output

1

Unity feedback

Step input

Vin=10

PD(s)

PID Controller

Td.s

(Td/N)s+1

Filter1

1

(Td/N)s+1

Filter

du/dt

=

Kp

.3

1

.2

..9

..8

..7

..6

0

..5

..4

..3

..2

..1

Td

.......

du/dt

.....

0

..'1

0

..'

0

..

du/dt

.'

Kp

.

Td

,,

1

s

,'2

,'1

,'

Kd

,

1/Ti

'1

'.

1

s

''

Ki

'

sy1.mat

0 5 10 15 200

5

10

15

20

Time (seconds)

Magnitude

sys1 P-design α=1

0 5 10 15 200

5

10

15

20

Time (seconds)

Magnitude

sys1 P-design α=2



Vol.3, No.7, 2013

11

Figure 7(a) P-controller design for system 2,3,4

3.2 PD-controller design for second order systems

Based on plant's damping and undamped natural frequency, expression given by Eq.(28) are proposed to design

PD-controller for second order systems. For systems with small time constant and/or small DC gain, expression

given by Eq.(29) are proposed, in both cases, only one tuning parameter α is introduced, that can be softly tuned

to speed up response, reduce overshoot and steady state error, where increasing parameter α will result in

decreasing overshoot, oscillation and reducing steady state error.

Applying PD controller design for system 1 for achieving desired output of 10, and by tuning parameter α to

have the values α =1,5,15, will result in response curves shown in Figure 8(a), the response curves show,

increasing α will result in reducing steady state error and overshoot, while meeting acceptable response.

Applying PD-controller design for achieving desired output of 10, for system 2 , will result in response curves

shown in Figure 8(b). Applying PD-controller design for achieving desired output of 10, for system 3 and

applying expression given by Eq.(29) will result in response curves shown in Figure 8(c). These response curves

show that a response with minimum overshoot and minimum error and with suitable settling time is achieved,

also by soft tuning of only one parameter α, the response can be softened .

2 2

2.9

2.92.9

P

n

D n

D

I n

P

n

RK

K R

K RT

RK

αξω

α ξω

α ξωξ ω

αξω

=

=

= = =

(28)

2

2

2 2

P

n

D n

D

I n

P

n

RK

K R

K RT

RK

αξω

α ξω

α ξωξ ω

αξω

=

=

= = =

(29)

Figure 8 PD(a) controller design for system 1, for α=1,5,15

0 5 10 150

5

10

15

Time (seconds)

Magnitude

sys2 P-design α=1

0 10 200

5

10

15

20

Time (seconds)

Magnitude

sys3 P-design α=5

0 50

5

10

15

Time (seconds)

Magnitude

sys4 P-design α=2

0 1 2 30

5

10

Time (seconds)

Magnitude

sys1 PD-design α=1

0 1 2 30

5

10

Time (seconds)

Magnitude

sys1 PD-design α=5

0 1 2 30

5

10

Time (seconds)

Magnitude

sys1 PD-design α=15



Vol.3, No.7, 2013

12

Figure 8(b) PD controller design for system 2, for α=1,5,15

Figure 8(c) PD controller design for system 3, for α=1,5,15

3.3 PI-controller design for second order systems

To design PI-controller for second order systems, expression given by Eq.(30) are proposed, to speed up

response, only one tuning parameter α is introduced.. Applying PI-design for second order systems given by

Eq.(27), will result in response curves shown in Figure 9, the response curves show that an acceptable response

with zero steady sate error and without overshoot are achieved.

( )

10

,牋牋? 牋? .8 1

10

n

I

n

P I

P n I

I I

I n

K

K K where

K KT T

K

ξ ωξω

α α

αξωα

ξ ω

+=

= = ÷

= = ⇒ =+

(30)

Figure 9 PI-controller design for three second order systems

3.4 PID-controller design for second order systems

The general standard form of second order system, can be expressed in terms of damping ratio ζ and undamped

natural frequency ωn as given by Eq.( 31). Since PID transfer function is a second order system, it can also be

expressed in terms of damping ratio and undamped natural frequency as given by Eq.(31), the PID gains ; KP, KI,

KD, can be found in terms of plant's damping ratio and undamped natural frequency, as given by Eqs.(32)(33).

Assigning proportional gain, the value of unity, KP=1, and equating Eqs.(31)(32), to find integral gain KI, wil

result in Eqs.(34) that is used to find numerical values of PID gains given by Eq.(35), based on plant damping

ratio and undamped natural frequency, to soften the response in terms of reducing overshoot, steady state error

and speed up response a tuning parameters ε and α with proposed rages are introduced. 2 22

2 2

2( ) 牋牋牋牋 ( )

2

D n nn

PID

n n

K s sG s G s

ss s

ξω ωωξω ω

+ + = ⇔ =+ +

(31)

0 1 2 30

5

10

Time (seconds)

Magnitude

sys2 PD-design α=1

0 1 2 30

5

10

Time (seconds)

Magnitude

sys2 PD-design α=5

0 1 2 30

5

10

Time (seconds)

Magnitude


0 5 100

5

10

Time (seconds)

Magnitude

sys3 PD-design α=1

0 1 2 30

5

10

Time (seconds)

Magnitude

sys3 PD-design α=5

0 1 2 30

5

10

Time (seconds)

Magnitude


0 10 20 300

5

10

Time (seconds)

Magnitude

sys1 PI-design α=1

0 10 20 300

5

10

Time (seconds)

Magnitude

sys2 PI-design α=1

0 20 400

5

10

Time (seconds)

Magnitude

sys3 PI-design α=1



Vol.3, No.7, 2013

13

2

2

I I

n D

D n

K KK

Kω

ω= ⇒ = (32)

22

P P

n D

D n

K KK

Kξω

ξω= ⇒ = (33)

2

2 2

1

2 2 2 2

1

2 2

1

P I I n n

I I

n n nn n

P

D D

n n

P

K K KK K

KK K

K

ω ωξω ξω ξω ξω ω

ξω ξω

= ⇒ = ⇒ = ⇒ =

= ⇒ =

=

(34)

11牋牋?牋牋牋牋牋牋,牋牋牋?

2 2

n

P D I

n

K K Kω

ξω ξ= = = (34)

Since PID transfer function, can be expressed in terms of derivative time and integral time to have the form

given by Eq.(35) Where:I T /P IK K= , the integral time and /D D PT K K= , the derivative time

211

1 I D IPID P D P

I I

T T s T sG K T s K

T s T s

+ += + + =

(35)

/I P IK K T= , D P DK K T=

Based on this, the derived formulae for calculating PID controller gains in terms of derivate time TD and integral

time TI , to be as given by Eq.(36), the divisor N is chosen in the range 2 to 20.

I

2 2T

/ 2 牋 ?

1/ 2 1

2

P P P

I n n n

D n

D D

P P n

K K K

K

KT K

K K

ξ ξω ξ ω ω

ξωξω

= = = =

= = = =

(36)

For soft tuning of proposed PID controller design, two tuning parameters are introduced α and ε , Where:

Increasing ε will increase overshoot, and vise versa, while increasing α will speed response, as a result, tuning

these parameters will result in reducing overshoot, reducing steady state error and speeding up response. The

proposed expressions for PID controller design are listed in Table 7.

Testing proposed PID controller design method for second order systems given by Eq.(27), will result in PID

terms, given in Table 8 , and response curves shown in Figure 10. These response curves show an acceptable

response without overshoot, steady state error and in acceptable settling time are achieved.

Further testing PID controller design procedure for second order systems given by Eq.(37), will result in PID

terms, tuned parameters values given in Table 9 , and response curves shown in Figure 11 .

2 2 2

1 1 11 ,牋牋牋牋 2 牋 ,牋牋牋牋牋 3 牋

1 6 5 10 10 40sys sys sys

s s s s s s= = =

+ + + + + + (37)

Table 7 Proposed expressions for PID gains calculation

Plant PID parameters

KP KI KD TD TI N

ζ ωn 1

2

nωξ

1

2n

ξω

1

2 nξω 2

n

ξω

2 20÷

Tuning

limits

1 ,牋? .1 10

2

nωε εξ

= ÷ 1

?牋? .58 1.52

n

α αξω

= ÷

2n

αξω

2

n

ξεω

Table 8 Testing results of second order systems

System Plant parameter PID parameters

parameter ζ ωn ε α KP KI KD

Sys(1) 0.5 1 0.62 1.1 1 0.62 1.1

Sys(2) 0.3536 1.4142 0.68 1.5 1 1.3598 1.4998

Sys(3) 0.7071 0.7071 8 1 1 4 1



Vol.3, No.7, 2013

14

Figure 10applying PID control method for three second order systems

Table 9 Testing results of second order systems

System Plant parameter PID parameters

parameter ζ ωn KP KI KD

Sys(1) 0.5 1 1 1 1

Sys(1), tuned only KI value 1 0.6300 1

Sys(2) 1.3416 2.2361 1 0.8333 0.1667

Sys(2), tuned only KI value 1 1.6667 0.1667

Sys(3) 0.2500 2 1 4 1

Sys(2), tuned only KI value 1 8 1

(a) Applying calculated PID gains (a) after tuning only KI

Figure 11 System (1) responses applying proposed PID design,


Figure 11 System (2) response applying proposed PID design.


Figure 11 System (3) response applying proposed PID design

0 5 100

5

10

Time (seconds)

Magnitude

sys1 PID, α=1.1 ε= 0.62

0 5 100

5

10

Time (seconds)

Magnitude

sys2 PID, α=1.5 ε= 0.68

0 5 10 150

5

10

Time (seconds)

Magnitude

sys3 PID, α=1 ε= 8

0 5 100

5

10

15

Time (seconds)

Magnitude

Sys(1) tuned PID, Ki*0.63

0 5 10 150

5

10

15

Time (seconds)

Magnitude

Sys(1) PID resp.

0 10 20 30 400

5

10

Time (seconds)

Magnitude

Sys(2)PID response

0 5 10 15 200

5

10

Time (seconds)

Magnitude

Sys(1) tuned PID, Ki*2

0 20 40 60 800

5

10

15

Time (seconds)

Magnitude

Sys(2)PID response

0 10 20 300

5

10

15

Time (seconds)

Magnitude

Sys(1) tuned PID, Ki*4

Industrial Engineering Letters

ISSN 2224-6096 (Paper) ISSN 2225-0581 (online

Vol.3, No.7, 2013

3.5 Summary: Controller design for second order systemsThe proposed method for second order systems and expressions for controllers terms

summarized in Table 10

Table 10 P, PI , PD, PID controllers terms for

Controller

type

KP

P-controller nRα ωξ

For systems

with small

DC gain

and/or small

T

2

nRα ωξ

PD

n

Rαξω

Systems with

small DC

gain and/or

small T

2

n

Rαξω

PI 0.8 2

IKα

α = ÷

PID 1 ,

2

nωε εξ

4. PID Controller for first order plus delay time

A large number of industrial plants can approximately be modeled by a first order plus time delay (FOPTD)

(Katsuhiko Ogata, 2010)(Saeed Tavakoli et al, 2003)

model with dead-time ,it transfer function is given by Eq.(38) and

s-shape curve with no overshoot is called

and time constant T, these two constants can be determined by drawing a tangent line at the infl

the s-shaped curve, and finding the intersection of the tangent line with time axis and steady state level

Figure 12), then the transfer function of these

transport lag and given by Eq.(38)[]:

( )

( ) 1

LsC s Ke

R s Ts

−

=+

Figure 12 s-shaped curve with terminology (Farhan A. Salem, 2013)

Based on plant's delay time L, time constant T, and steady state level

0581 (online)

15

3.5 Summary: Controller design for second order systems The proposed method for second order systems and expressions for controllers terms selection and design are

Table 10 P, PI , PD, PID controllers terms for second order systems

KI KD TD

0

0

0

0 2.9* nRα ξω 2 22.9 nξ ω

0 2

nRα ξω 2 2

nξ ω

10

n

n

ξ ωξω+

0

0 I

I

T

T

,牋? .1 22

nωε εξ

= ÷1

?牋2

? .58 1.5

n

αξω

α = ÷

2 n

αξω

εω

4. PID Controller for first order plus delay time ( FOPDT) process

lants can approximately be modeled by a first order plus time delay (FOPTD)

(Katsuhiko Ogata, 2010)(Saeed Tavakoli et al, 2003). FOPDT models are a combination of a first

transfer function is given by Eq.(38) and it response curve is shown in Figure 12, this

is called reaction curve, it is characterized by two constants ; the delay time

, these two constants can be determined by drawing a tangent line at the infl

shaped curve, and finding the intersection of the tangent line with time axis and steady state level

Figure 12), then the transfer function of these-shaped curve can be approximated by first order system with

nd given by Eq.(38)[]:

shaped curve with terminology (Farhan A. Salem, 2013)

Based on plant's delay time L, time constant T, and steady state level K, ( see Eq. (12)). The formulae listed in

www.iiste.org

selection and design are

TI N

0

0

0 1-22

0 1-22

( )10

P

I

I

n I

I

n

KT

K

KT

α

αξωξ ω

= = ⇒

=+

1-22

2

n

ξεω

1-22

lants can approximately be modeled by a first order plus time delay (FOPTD)

. FOPDT models are a combination of a first-order process

esponse curve is shown in Figure 12, this

is characterized by two constants ; the delay time L,

, these two constants can be determined by drawing a tangent line at the inflection point of

shaped curve, and finding the intersection of the tangent line with time axis and steady state level K, (see

shaped curve can be approximated by first order system with

(38)

shaped curve with terminology (Farhan A. Salem, 2013)

, ( see Eq. (12)). The formulae listed in



Vol.3, No.7, 2013

16

Table 11, are proposed to calculate PID gains in terms L, T, K for ( FOPDT) process and tuning limits for KD and

KI. Based on Eqs.(35)(36), the derived formulae for calculating PID controller gains in terms of derivate time TD

and integral time TI , to be as given by in Table 11, the divisor N is chosen in the range 2 to 20.

Table 11 Proposed formulae for PID gains calculation for first ( FOPDT) system, and softening ranges

Plant PID parameters

KP KI KD TI TD N

T,K,L T *L T L

T 1 / L

2

L

T 2 20÷

Tuning

limits

T * * ,

0.1 3

L Tεε = ÷

0.1 3

L

Tα

α = ÷

1

*Lε

2

L

Tα

Applying proposed PID design method for first-order process with dead-time given by Eq.(39), will result in

the calculated PID gains values listed in Table 12, the response curve is shown in figure 13 0.3( )

( ) 1

sC s e

R s s

−

=+

(39)

Table 12 design for FOPDT

System Plant's parameters PID parameters

FOPDT

Parameters α ε KP KI KD TD TI N

L=0.3,T=1,K=1 2 2 1 0.60 0.6 6.6666 0.6 2

(a) Applying calculated PID gains (a) after tuning KD and KI

Figure 13 FOPDT response applying proposed PID design,

5. Comparing and testing proposed design with existing design methods

Case (1): Considering a third order plant with transfer function given by Eq.(40), to verify proposed design, it

will be compared with Ziegler-Nicols design method

3

1( )

( 1)G s

s=

+ (40)

Since this is third order system, it can be approximated as second order system with two repeated pole P=1,

Correspondingly ζ=1, ω =1, designing P, PI, PD, and PID controller applying Ziegler-Nicols design method and

proposed design will result in gains values listed in table 13, and the response curves are shown in Figure 14 ,

these response curves show that the proposed method is more simpler than Ziegler-Nicols, as well as a more

smooth response with minimum overshoot and acceptable settling time is achieved

Table 13 controllers' gains values applying both Ziegler-Nicols and proposed design methods

Parameters Pro. PD PI

PID

Proposed

method

ζ ω KP KP KD KP KI KP KI KD

1 1 5 20 58 0.1 0.1 1 0.5 0.5

Ziegler-Nicols

method

Kcrit Pcrit KP KP TD KP TI KP TI TD

8 3.62 4.8 - - 3.6 3 4.8 1.8 0.45

0 10 20 30 400

5

10

15

Time (seconds)

Magnitude

sys (4) PID design

0 5 10 15 200

5

10

15

Time (seconds)

Magnitude

sys (2) PID with Kd*2 ,Ki*2



Vol.3, No.7, 2013

17

Figure 14(a) P-controller design applying Ziegler-Nicols and proposed design methods

Figure 14(b) PD-controller design applying proposed design methods

Figure 14(c) PI-controller design applying Ziegler-Nicols and proposed design methods

Figure 14(c) PID-controller design applying Ziegler-Nicols and proposed design methods

0 10 20 30 40 500

2

4

6

8

10

12

14

Magnitude

Comparison P-controler design

Ziegler-Nicols method

Proposed method

0 50 1000

5

10

15

Time (seconds)

Magnitude

Comparison: PD-controler design

Proposed method

0 10 20 30 40 50 60 700

2

4

6

8

10

12

14

16

Magnitude

Comparison: PI-controler design

Proposed method

Ziegler nicols

0 5 10 15 20 250

2

4

6

8

10

12

14

16

Magnitude

Comparison: PID-controler design

Proposed method

Ziegler-Nicols method



Vol.3, No.7, 2013

18

Case (2): Testing proposed PID design method for fourth order plant transfer function given by Eq.(41),

applying three different PID controller design methods, particularly, Ziegler Nichols frequency response,

Ziegler-Nichols step response, and Chein-Hrones-Reswick design methods, will result in PID gains shown in

Table 14(Robert A. Paz, 2001), , as shown in this table different values of PID gain are obtained and

correspondingly different system's responses (see figure 15), when subjected to step input of 10. Comparing

shown response curves, show that the Chein-Hrones-Reswick design is, with less overshoot and oscillation (than

Ziegler-Nicols), all three method allmostly, result in the same settling time, Applying the proposed method,

based on plant's dominant poles approximation, result in smooth response curve without overshoot, and zero

steady state error, shown in figure 14.

4 3 2 s

10000( )

s + 126s + 2725s + 12600 + 10000G s = (41)

Table 14

Design Method KP KI KD

Ziegler Nichols Frequency Response 14.496 45.300 1.1597

Ziegler-Nichols Step Response 11.1524 34.3786 0.9045

Chein-Hrones-Reswick 5.5762 5.0794 0.4522

Proposed method

1 0.8632 0.1231

ζ=3.0677 ωn= 2.6481

Figure 15 System step responses obtained applying different design methodologies

Conclusion

A new simple and efficient model-based time domain P, PI ,PD , and PID controllers design method for

achieving an important design compromise; acceptable stability, and medium fastness of response is proposed,

the proposed method is based on selecting controllers' gains based on plant's parameters, the proposed controllers

design method was test for first, second and first order system with time delay, and using MATLAB/simulink

software.

References Ahmad A. Mahfouz, Mohammed M. K., Farhan A. Salem, "Modeling, Simulation and Dynamics Analysis Issues

of Electric Motor, for Mechatronics Applications, Using Different Approaches and Verification by

MATLAB/Simulink", IJISA, vol.5, no.5, pp.39-57, 2013.

Astrom K,J, T. Hagllund, PID controllers Theory, Design and Tuning , 2nd edition, Instrument Society of

America,1994

Ashish Tewari, Modern Control Design with MATLAB and Simulink, John Wiley and sons, LTD, 2002 England

Katsuhiko Ogata, modern control engineering, third edition, Prentice hall, 1997

0 1 2 3 4 5 6 70

2

4

6

8

10

12

14

16

18

Response applying different PID design methodologies

Time (seconds) (sec)

Ma

gnitu

de

Open loop response

Ziegler-Nicols step response method

Chein Hrones Reswick Method

Proposed method



Vol.3, No.7, 2013

19

Farid Golnaraghi Benjamin C.Kuo, Automatic Control Systems, John Wiley and sons INC .2010

Norman S. Nise, Control system engineering, Sixth Edition John Wiley & Sons, Inc,2011

Gene F. Franklin, J. David Powell, and Abbas Emami-Naeini, Feedback Control of Dynamic Systems, 4th Ed.,

Prentice Hall, 2002.

Dale E. Seborg, Thomas F. Edgar, Duncan A. Mellichamp ,Process dynamics and control, Second edition, Wiley

2004

Dingyu Xue, YangQuan Chen, and Derek P. Atherton "Linear Feedback Control". 2007 by the Society for

Industrial and Applied Mathematics, Society for Industrial and Applied Mathematics the Society for Industrial

and Applied Mathematics, 2007

Chen C.L, A Simple Method for Online Identification and Controller Tuning , AIChe J,35,2037 ,1989.

Lee J., Online PID Controller Tuning For A Single Closed Test, AIChe J,32(2), 1989.

R. Matousek, HC12: Efficient PID Controller Design, Engineering Letters, pp 41-48, 20:1, 2012

K. J. Astrom and T. Hagglund, The Future of PID Control, IFAC J. Control Engineering Practice, Vol. 9, 2001.

Susmita Das, Ayan Chakraborty,, Jayanta Kumar Ray, Soumyendu Bhattacharjee. Biswarup Neogi, Study on

Different Tuning Approach with Incorporation of Simulation Aspect for Z-N (Ziegler-Nichols) Rules,

International Journal of Scientific and Research Publications, Volume 2, Issue 8, August 2012

L. Ntogramatzidis, A. Ferrante, Exact tuning of PID controllers in control feedback design, IET Control Theory

and Applications, 2010.

M.Saranya , D.Pamela , A Real Time IMC Tuned PID Controller for DC Motor design is introduced and

implemented, International Journal of Recent Technology and Engineering (IJRTE), Volume-1, Issue-1, April

2012

Fernando G. Martons, Tuning PID controllers using the ITAE criterion, Int. J. Engng Ed. Vol 21, No 3 pp.000-

000-2005 .

Saeed Tavakoli, Mahdi Tavakoli, optimal tuning of PID controllers for first order plus time delay models using

dimensional analysis, The Fourth International Conference on Control and Automation (ICCA’03), 10-12 June

2003, Montreal, Canada

Farhan A. Salem, Controllers and control algorithms; selection and time domain design techniques applied in

mechatronics systems design; Review and Research (I) , submitted to international journal of engineering

science , 2013.

Farhan A. Salem, Precise Performance Measures for Mechatronics Systems, Verified and Supported by New

MATLAB Built-in Function', International Journal of Current Engineering and Technology, Vol.3, No.2 (June

2013).

Farhan A. Salem, PID Controller and algorithms; selection and design techniques applied in mechatronics

systems design (II) ,submitted to International Journal of Engineering Sciences, 2013.

Robert A. Paz , The Design of the PID Controller, Klipsch School of Electrical and Computer Engineering,

2001.

Pradeep Kumar Juneja , A. K. Ray, R. Mitra, Deadtime Modeling for First Order Plus Dead Time Process in a

Process Industry, International Journal of Computer Science & CommunicationVol. 1, No. 2, July-December

2010, pp. 167-169.

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